1. **Problem statement:** Find the angle $\angle XWZ$ in trapezium XYZW where $XY=7$, $WY=11$, $WX=8$, and $YZ=$ unknown but not needed. 2. **Understanding the trapezium:** Points are labeled as $X, Y, Z, W$ with given side lengths. We want $\angle XWZ$, the angle at vertex $W$ formed by points $X$ and $Z$. 3. **Approach:** Use the cosine rule in triangle $WXZ$ to find $\angle XWZ$. 4. **Find length $XZ$:** Since $XY=7$, $YZ=11$, and $WX=8$, and $WXYZ$ is a trapezium, we can find $XZ$ using the Pythagorean theorem or coordinate geometry. But here, we use the given sides to find $XZ$. 5. **Calculate $XZ$:** Using the trapezium properties, $XZ$ is the diagonal opposite to $W$. We can use the triangle $XYZ$ or $WXZ$. Since $YZ=11$ and $XY=7$, and $WX=8$, we consider triangle $WXZ$ with sides $WX=8$, $XZ=$ unknown, and $WZ=11$. 6. **Apply cosine rule in triangle $WXZ$:** $$ \cos(\angle XWZ) = \frac{WX^2 + WZ^2 - XZ^2}{2 \times WX \times WZ} $$ 7. **Calculate $XZ$:** Using the trapezium, $XZ = \sqrt{XY^2 + YZ^2} = \sqrt{7^2 + 11^2} = \sqrt{49 + 121} = \sqrt{170} \approx 13.0384$. 8. **Substitute values:** $$ \cos(\angle XWZ) = \frac{8^2 + 11^2 - (13.0384)^2}{2 \times 8 \times 11} = \frac{64 + 121 - 170}{176} = \frac{15}{176} \approx 0.0852 $$ 9. **Find angle:** $$ \angle XWZ = \cos^{-1}(0.0852) \approx 85.11^\circ $$ 10. **Final answer:** $\boxed{85.11^\circ}$ is the measure of $\angle XWZ$.